types ideals on is-algebras - m- sundus najah jabir definition 3.3 let k and h be an ideals in...

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  • International Journal of Mathematical Analysis

    Vol. 11, 2017, no. 13, 635 - 646

    HIKARI Ltd, www.m-hikari.com

    https://doi.org/10.12988/ijma.2017.7466

    Types Ideals on IS-Algebras

    Sundus Najah Jabir

    Faculty of Education

    Kufa University, Iraq

    Copyright 2017 Sundus Najah Jabir. This article is distributed under the Creative Commons

    Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

    provided the original work is properly cited.

    Abstract

    In this paper we study I-ideals and P-ideals on IS-algebra and prove some

    results about this.

    Keywords: BCI- algebras, semigroup, IS-algebra, I-ideals, P-ideals

    1 Introduction

    The class of BCI- algebras which was introduced by K.Iseki [1] in 1966 is an important class of logical algebras which has two origins.one of the

    motivations for their use is based on set theory, the other on classical and non-

    classical propositional calculus.by definition, [6] introduced a new class of

    algebras related to BCI- algebras and semigroups called a BCI- semigroup. From

    now on, we rename it as an IS- algebra for the convenience of study.

    2 Preliminary

    We review some definitions and properties that will be useful in our results.

    Definition 2.1 A Semigroup is an ordered pair ),( G , where G is a non empty set

    and " ." is an associative binary operation on G. [3]

    Definition 2.2 A BCI- algebra is triple (G ,* ,0) where G is a non empty set "*" is

    binary operation on G , G0 is an element such that the following axioms are

    satisfied for all Grts ,, :

    1) ((s t) (s r)) (r t) = 0, 2) (s (s t) t = 0,

  • 636 Sundus Najah Jabir

    3) s s = 0, 4) s t = 0 and t s = 0 implies s = t If 0 s = 0 for all Gs then G is called BCK-algebra. [12]

    Definition 2.3 An IS-algebra is a non empty set with two binary operation "*"

    and "." and constant 0 satisfying the axioms:

    1. (G , 0) is a BCI-algebra. 2. (G, .) is a Semigroup,

    3. s.(t r) = (s.t (s.r) and (s t).r = (s.r) (t.r), for all s, t, r G . [9]

    Example 2.4 let G={0,n,m,v} define "*" operation and multiplication "." by the following tables:

    . 0 n m v

    0 0 0 0 0

    n 0 n 0 n

    m 0 0 m m

    v 0 0 m v

    Then by routine calculations we can see that G is an IS-algebra.[9]

    Example 2.5 let G={0,n,m,v,u} define "*" operation and multiplication "." by the

    following tables:

    . 0 n m v u

    0 0 0 0 0 0

    n 0 0 0 0 0

    m 0 0 0 0 m

    v 0 0 0 m v

    u 0 n m v u

    Then G is an IS-algebra. [9]

    * 0 n m v

    0 0 0 v m

    n n 0 c m

    m m m 0 v

    v v v m 0

    * 0 n m v u

    0 0 0 0 0 0

    n n 0 n n 0

    m m m 0 0 0

    v v v v 0 0

    u u u u u 0

  • Types ideals on IS-algebras 637

    Definition 2.6 A non- empty subset K of a semigroup G is said to be left (resp.

    right ) stable if ).( KnsrespKsn whenever KnandGs both

    left and right stable is two sided stable or simply stable. [9]

    Definition 2.7 A non- empty subset K of IS-algebra G is called a left (resp. right)

    I-ideal of G if:

    1) K is a left (resp. right) stable of G.

    2) For any KsthatimplyKtandKtsGts *,, .

    Both left and right I-ideal is called a two sided I-ideal or simply an I-ideal. [9]

    Definition 2.8 A binary relation on G by letting ts

    if and only if

    0* ts is a partial ordered set . [10]

    3 Main Results

    In this section, we find some results about I-ideals and P-ideals on IS-algebra.

    Proposition 3.1 Let K and H are left (resp. right) I-ideals of G Then HK is a left (resp. right) I-ideals of G.

    Proof: Let K and H be left I-ideals of G ,

    and let Gs and HKn Then HnandKn

    HsnandKsnso [since K , H are left I-ideals ]

    HKsnthen

    HKtandHKtsletNow *, HtKtandHtsKtsthen ,*,*

    HsandKsso [since K , H are left I-ideals]

    HKstherefore

    HKHence is a left I-ideal.

    Proposition 3.2 Let K and H are left (resp. right) I-ideals of G Then HK is a left (resp. right) I-ideals KHorHKIf .

    Proof: Suppose that K and H are left I-ideals of G

    Without loss of generality we may assume that

    HK then HHK since H is a left I-ideal

    so HK is a left I-ideal.

  • 638 Sundus Najah Jabir

    Definition 3.3 let K and H be an ideals in IS-algebra Then

    },:),{( HmKnmnHK is an ideal where the binary operations "" and " * " are define by the following:

    ),(),(),( 21212211 mmnnmnmn ,

    )*,*(),(*),( 21212211 mmnnmnmn for all HKmnmn ),(),,( 2211 .

    Proposition 3.4

    Let K and H be a left (resp. right ) I-ideals of IS-semigroups G . Then HK is a left (resp. right) I-ideal of GG .

    Proof:

    Let K and H are left I-ideal of IS-semigroups G

    Let HKnnGGss ),(,),( 2121 then

    ,

    ),(),(

    ),(

    ],[sin

    ),(),(),(

    2121

    21

    2211

    22112121

    11

    Now

    HKnnssso

    HKnsnsThen

    idealsIleftareHKceHnsandKnsSince

    nsnsnnss

    HKsthen

    HKssso

    idealsIleftareHKceHsandKsthen

    HtKtandHtsKtsthen

    HKttandHKtststhen

    HKttandHKttssif

    GGtttsss

    whereHKtandHKtslet

    ),(

    ],[sin

    ,*,*

    ),()*,*(

    ),(),(*),(

    ),(,),(

    ,)*(

    21

    21

    212211

    212211

    212121

    2121

    Hence HK is a left I-ideal.

    Definition 3.5 Let G and R be IS-algebra a mapping RG : is called a IS-

    algebra homomorphism (briefly homomorphism) if )(*)()*( tsts and

    )()()( tsts for all Gts , .

    Let RG : IS-algebra homomorphism. Then the set }0)(:{ sGs is

    called the kernel of , and denote by ker . Moreover, the set

    }:)({ GsRs is called the image of and denote by Im .

  • Types ideals on IS-algebras 639

    Definition 3.6 Let G and R be an IS-algebra such that RG : IS-algebra

    homomorphism then :

    1) is a monomorphism iff one to one homomorphism .

    2) is an epimorphism iff onto homomorphism .

    3) is an isomorphism iff bijective homomorphism .

    Proposition 3.7 Let RG : be an IS-semigroup homomorphism Then ker () is a I-ideal of G.

    Proof:

    Let RG : be a IS-semigroup homomorphism,

    to prove ker ( ) is a stable

    let )(ker nandGs so 0)( n then

    00).()()()( snsns [since is a homomorphism]

    thus kersn

    then ker ( ) is a stable .

    Now,

    let Gts , such that

    kerker* tandts ,

    so 0)(0)*( tandts

    so 0)(0)(*)( tandts [since is a homomorphism]

    then 00*)( s [since kert ]

    thus 0)( s then kers

    Hence ker ( ) is a I- ideal.

    Proposition 3.8 Let RG : be an IS-semigroup epimorphism and let K is a

    left (resp. right) I-ideal in G .Then )(K is a left (resp. right) I-ideal in R.

    Proof:

    Let K be a left I-ideal of G

    let KnwhereRtandKnn )()(

    since onto then there exists tsthatsuchGs )(

  • 640 Sundus Najah Jabir

    To prove )(Ktn

    since KnandGsidealIleftaisKceKsn ][sin

    so )()( Ksn

    but tnnsns )()()(

    [ is epimorphism ]

    therefore )(K is stable .

    Now, suppose that KtssomeforKts ,)()(),( ,

    such that )()()()(*)( KtandKts

    To prove )()( Ks

    since is a homomorphism then

    )(*)( ts ,)()(sin)()*( KtceandKts

    thus KsKtKts ,* [since K is I-ideal]

    therefore )()( Ks

    Hence )(K is a left I-ideal in R .

    Theorem 3.9 Let RG : be an IS-semigroup homomorphism if K is an I-

    ideal of R then })(/{)(1 KnGnK is an I-ideal of G containing ker

    ( ) .

    Proof:

    Let RG : be an IS-semigroup homomorphism

    Suppose that K is an I-ideal of R

    Let )(1 KrandGs then Kr )( and

    KsrrsandKrssr )()()()()()(

    Since K is stable and )(1, Krssr

    Thus )(1 K is stable.

    Now,

  • Types ideals on IS-algebras 641

    )(1)(1*, KtandKtsthatsuchGts then

    )(*)( ts KtandKts )()*(

    Then Ks )( [Since K is an I-ideal ]

    That is )(1 Ks therefore )(1 K is an I-ideal of G.

    Moreover K}0{ implies that ker ( )(1})0({1) K .

    Definition 3.10 Let K and H be ideals of IS-algebra G define

    },:{ HmKnKHnmKH .

    Lemma 3.11 Let K and H be an I-ideals of IS-semigroup G with unity such that

    HK 1 Then KH is an I-ideal of G.

    Proof:

    Assume that K and H be an I-ideals

    Let KHnmandGs then

    KHmsnsnm )()( and KHmsnnms )()( [since

    KsnHsm , ]

    Thus KH is stable subset of G .

    Now,

    Suppose that Gts , such that KHtandKHts * then

    nmstsandnmt ** since K and H be an I-ideals and

    HnmKnm ,